recombinant estimation for normal-form games, with ... · recombinant estimation for normal-form...

Recombinant Estimation for Normal-Form Games,

with Applications to Auctions and Bargaining

Charles H. Mullin*

David H. Reiley†*

This Version: November 2004 First Version: January 1999

Abstract When analyzing economic games, researchers frequently estimate quantities describing group outcomes, such as the expected revenue in an auction. For such applications, we propose an improved statistical estimation technique called “recombinant estimation.” The technique takes observations of players’ strategies and recombines them to compute all possible group outcomes that could have resulted from different player combinations. In applications to an auction and a bargaining game, the improved efficiency of our estimator is equivalent to increasing the sample size between 25% and 200%. We discuss how to design experiments in order to utilize recombinant estimation. We also discuss practical computational issues, and provide software for computing estimates and standard errors. JEL Classification: C70, C13 and C92. Keywords: Simultaneous-Move Games, Experimental Statistics

* Mullin can be contacted at [email protected], and Reiley can be reached at [email protected]. We thank J.S. Butler, Yan Chen, Rachel Croson, Rosemarie Nagel, Karim Sadrieh, and Mark Satterthwaite for valuable discussions. Mike Urbancic has done a wonderful job writing flexible, user-friendly software in Microsoft Excel, so that others may easily use our estimator without tedious programming. We are also grateful for the comments of an anonymous referee and an associate editor. † Reiley gratefully acknowledges the National Science Foundation for support through grants SBR-9811273 and SES-0094800.

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1. Introduction

Economists frequently study behavior in simultaneous-move games with multiple players, from

auctions to Bertrand oligopoly to public-good provision. Recent interest in behavioral game

theory has generated a wealth of experimental data on individuals’ play in a wide variety of such

games. These data can be used to make inferences about various quantities of economic interest,

such as auction revenues, market prices, and levels of public-good provision. Often, experiments

on multi-player games group each subject randomly and arbitrarily with anonymous opponents.

In such simultaneous-move games with arbitrary groupings, it seems reasonable to assume that

the identities of any given player’s rivals do not alter her strategy. This assumption implies that

the hypothetical game outcomes which would result from grouping a subject with players she did

not actually face are just as valid as the game outcome resulting from the group of players she

did play against. In this paper, we develop an estimation technique, which we call “recombinant

estimation,” designed to extract maximal information from observed strategies in order to

estimate various properties of simultaneous-move games.

For concreteness, we present the example that originally motivated this paper. List and

Lucking-Reiley (2000) compare two different sealed-bid auction formats. They conducted 15

identical two-person, uniform-price auctions for identical pairs of goods. This yielded data from

30 bidders, all of whom faced the identical bidding situation. They repeated this procedure with

30 additional bidders in 15 different multi-unit Vickrey auctions. To compare the two auction

formats, they conducted t-tests, which involve estimating both the mean and the standard error of

the mean for each auction format. Recall that with n independent observations, the standard error

of the mean equals the standard deviation of the data divided by the square root of n. In order to

test whether the mean bids were equal across auction formats, they computed the mean bid, and

computed the standard error as the standard deviation divided by the square root of 30. They also

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compared revenues across auction formats, but since there were only 15 revenue observations

(15 auctions with 2 bidders each), the standard error for this case was equal to that standard

deviation divided only by the square root of 15. This seemed somehow unfair, as the same

amount of information led to relatively higher standard errors for revenues than for bids.

This observation led us to realize that in a sealed-bid auction, the pairings of bidders were

arbitrary, so we could observe the revenues that would have resulted from all 435 possible

pairings (the number of combinations of 30 bidders taken two at a time). Computing the mean

over these 435 different revenue observation clearly gives a more precise estimate of the mean

revenue in the population of auctions with all possible bidders, since it uses more information.

But what should be the standard error of the mean? Surely one can’t just divide the standard

deviation by the square root of 435; there’s not that much information available! In this paper,

we formalize the intuition just discussed, presenting a recombinant estimator for use in a variety

of normal-form game applications, and deriving correct standard-error estimates so that the

estimator may be used for inference purposes.

We are not the first to propose recombining players to attain more-accurate estimators.

For example, in their analysis of a bargaining game, Mitzkewitz and Nagel (1993) use the

complete contingent strategies for each player to compute recombinant estimates of expected

profits. Similarly, Mehta et al. (1994) utilize recombinant estimation to determine the frequency

of success in pure coordination games. Verifying the intuition of this approach, Friedman (1996)

shows that players learn a game faster if they receive the expected payoff across all possible

opponents, as opposed to the payoff based solely on the actions of one randomly chosen

opponent.

What is new in this paper is the placement of the idea of recombinant estimation into a

formal econometric framework. First and foremost, we derive standard errors for this estimation

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technique, which both quantifies the efficiency improvement associated with this technique and

enables hypothesis testing. For example, in the motivating example of List and Lucking-Reiley

(2000), we attain efficiency improvements of 19 to 68 percent, depending on the quantity

estimated, which is roughly equivalent to increasing the sample size by 25 to 200 percent. When

applied to the Mitzkewitz and Nagel (1993) data, the recombinant estimator improves efficiency

16 to 68 percent, with the efficiency gains increasing with the complexity of the strategy space.

Second, formalizing the estimation technique clearly demonstrates when it can and

cannot be used, which translates into some recommendations for experimental design. The key is

to make sure that players face the same public environment in each session of the experiment

(e.g., the number of players, the structure of the payoff function, the distribution of information,

etc.), so that the recombination across sessions is meaningful. In other words, the only items

allowed to change across sessions are the actual players taking each role and the realization of

any information that is private to a particular player.

Third, recombinant estimation can result in a combinatorial explosion in the number of

computations required. Depending on the number of players per group, a few dozen actual

observations might spawn billions of hypothetical observations. We show that one can achieve

almost all of the efficiency gains of recombinant estimation by considering only a random

sample of several thousand of the hypothetical observations, an estimation task readily achieved

by modern computers in seconds or less. Similar practical considerations apply to the estimation

of standard errors for the estimator, where the combinatorial explosions are even more rapid than

in the computation of the estimator itself. Using these practical insights, we have implemented

our estimator in user-friendly computer software, available for download from the second

author’s Web site. 1

1 The software is written with macros in Microsoft Excel. To use it, one copies one’s data into the Excel workbook, enters the group outcome formula, and executes the software to obtain estimates of the mean group outcome and the standard error of this mean. At this writing, the software works only for symmetric games, but we hope in future to

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Finally, recombinant estimation bears some similarities to the statistical technique of

bootstrapping, so it may be worth pointing out the differences. A traditional bootstrap reuses the

data in a similar manner, but they do so in order to attain more-robust confidence intervals for

the test statistic of interest. More specifically, bootstrapping provides a two-step procedure to

estimate numerically the sampling distribution of an estimator. The bootstrap first estimates the

data-generating process from the empirical distribution of the data, and then uses Monte Carlo

simulations based on this estimated data-generating process to estimate the sampling distribution

of the estimator. Thus, while bootstrapping is a method for doing inference with an existing

estimator, recombinant estimation constitutes a new, more efficient estimator for a population

parameter.2 In other words, we switch the focus from inference about a given estimator to

developing a more efficient estimator.

The remainder of the paper is organized as follows: Section 2 develops the estimator and

its distribution. Section 3 addresses the efficiency of the estimator. Section 4 provides two

empirical examples, while Section 5 discusses how to design experiments in order to take

advantage of this estimation technique. Section 6 presents practical computational

considerations, and Section 7 concludes.

2. Estimators for Game Outcomes

In this section, we develop the recombinant estimator for the expected value of an outcome in a

simultaneous-move game. For the sake of exposition, we start with relatively simple games, and

add more generality in later subsections. In each case we consider, the estimator is a sample

average that satisfies a central limit theorem, so it has an asymptotically normal sampling

distribution.

provide software for asymmetric games as well. The software is currently available at <http://eller.arizona.edu/~reiley/papers/RecombinantGames.html>. 2 Similarly, the Fisher randomization test is solely a method for doing inference, not an estimation technique. See Moir (1998) for details.

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Before developing the estimator in detail, we introduce some notation. Assume that the

econometrician observes n groups of k players each, and let m = nk denote the total number of

players. Let xi be the strategy observed for the ith player and pj be the vector of players who

participated in group j. For concreteness, we might think of the first group of players as p1 = (1,

2, …, k), p2 = (k + 1, k + 2, …, 2k) as the second group, and so on. Denote pjt as the tth player in

group j. Define yj as the outcome which results from the strategies played by the set of players pj,

and let g be the function which maps strategies to group outcomes, so that

yj = g xpj1

, xpj 2,K , xpjk( ).

The goal is to estimate the expected value of y over the entire population of possible

players. Let y have a mean of µ and a variance of σ2. For comparison to the recombinant

estimator, we first present the baseline estimator most commonly observed in practice. This

baseline estimator computes the simple average of the group outcome for each of the n observed

groups. We denote this baseline estimator by:

µ = y = 1 n( ) yjj=1

n

∑ (2-1)

where j indexes the n groups. The variance of this estimator is

var y( )= σ 2 n . (2-2)

This estimator serves as our reference point for efficiency comparisons to other proposed

estimators.

In the following subsections, we first develop our recombinant estimator for symmetric

two-player games and then generalize it to symmetric k-player games. We next develop a

recombinant estimator for fully asymmetric k-player games, and finally consider general k-player

games. The special cases provide a simplified setting in which to explain the intuition behind the

recombinant estimation strategy.

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2.1 Symmetric Two-Player Games

Suppose we observe four players matched in two pairs in a two-player, symmetric game. Let y1

represent the outcome for players 1 and 2, and y2 represent the outcome for players 3 and 4.

Using the baseline estimator, the average of these two outcomes would be our estimate.

However, one might be able to improve upon this estimator by considering alternative

combinations of the players. In particular, the baseline estimator ignores the potential pairings

1,3,1,4,2,3,2,4.

The recombinant estimator considers these potential pairings. Recall that pj represents the

set of players in group j. For the baseline estimator, j indexed only those groups actually

observed, so it ranged from 1 to n. Now we let P represent the collection of all possible 2-player

sets which can be taken from the m players, and let j index all the elements of P, so that j now

ranges from 1 to J ≡ m!/(2!(m – 2)!). The recombinant estimator is the simple average of these J

combinations:

µ =1J

⎛⎝⎜

⎞⎠⎟

yjj=1

J

∑ . (2-3)

Note that the set P fails to distinguish between the pairings i, j and j, i. Due to the symmetry

in the outcome function, these two outcomes correlate perfectly, so the second adds no new

information once the first has been considered.

We have chosen to ignore the hypothetical outcomes in which a player is matched with

herself. In some applications, the researcher may find it appropriate to include additional terms

of the form i, i in the estimator of the mean. The additional information would improve the

efficiency of the estimator in small samples, but the effect is negligible in large samples, as the

number of i, i terms becomes negligible relative to the total number of i, j terms. The two

approaches are thus asymptotically equivalent, but the present form is notationally less

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burdensome. It is relatively straightforward to adapt the equations in this paper to include these

additional pairings.

To present the variance of this estimator requires some additional notation. Let ϕ ≡

cov(yi, yj) when groups i and j share exactly one common player. Note that since each player's

strategy is independent of all other players’ strategies, when there is no intersection between two

groups of players the covariance is zero. So,

var µ( )= 1 J( )2 var yj( )j=1

J

∑ + cov yj , yi( )i≠ j∑

j=1

J

∑⎡

⎣⎢

⎤

⎦⎥

= 1 J( )2 Jσ 2 + m m −1( ) m − 2( )ϕ⎡⎣ ⎤⎦

=2

m m −1( )σ 2 +

4 m − 2( )m m −1( )

ϕ

(2-4)

where m(m – 1)(m – 2) is the number of covariance terms with one common player across the

two groups. Recall that m = 2n. Therefore, when ϕ is positive, n ⋅ var µ( ) converges to 2ϕ.3 From

(2-2), n ⋅var y( ) converges to σ2. Thus, the asymptotic-efficiency gain of the recombinant

estimator relative to the baseline estimator is 2ϕ/σ2. Proposition 1 in the following section proves

that the recombinant estimator is weakly more efficient than the baseline estimator.

When ϕ is zero, converges to σn2 ⋅var µ( ) 2/2, which is a faster rate of convergence (n2

instead of n) than the baseline estimator. In general, whenever the covariance is zero, regardless

of the symmetry of the game or the number of players involved, the recombinant estimator

converges at a faster rate than the baseline estimator. Although this case is interesting, it is

atypical: for example, two hypothetical auctions with the same aggressive bidder will both tend

to have high revenues, and the opposite will be true for two hypothetical auctions with the same

conservative bidder. For that reason, from this point onward we concentrate on the case where ϕ

is strictly positive.

3 The average covariance across games with at least one shared player, ϕ, must be non-negative.

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For inference purposes, it is important both to be able to estimate the variance given by

(2-4) and to demonstrate that this variance fully describes the distribution of the estimator. First,

to estimate the variance, one can use all the hypothetical outcomes yj to produce consistent

estimators of σ2 and of ϕ, and then substitute the resultant estimates into (2-4). Since (2-4) gives

the exact equation for the variance, this estimate of the variance differs from the true variance

only insofar as the estimates of σ2 and ϕ differ from their population values. We will present

more details on computing standard errors in Section 6.2. For now, it is sufficient to note that the

recombinant estimation technique can be used to estimate both σ2 and ϕ, which increases the

accuracy with which these parameters are estimated. Second, the central limit theorem

establishes that the recombinant estimator follows a normal distribution. Furthermore, the

estimator is always a sample average (of repeated draws from a stationary distribution) and these

averages tend to converge to normal distributions relatively quickly.4

2.2 Symmetric K-Player Games

With more than two players in a symmetric game, there are many more recombinations possible.

Instead of merely m(m – 1)/2 different hypothetical outcomes from the m players, we now have

the number of combinations of m objects taken k at a time. Now we let P represent the collection

of all possible k-player sets which can be taken from the m players, and let j index all the

elements of P, so that j now ranges from 1 to J ≡ m!/(k!(m – k)!). Under this new definition of P,

the recombinant estimator is still given by (2-3), µ = 1 J( ) yjj=1

J

∑ .

To compute the asymptotic variance of this estimator, we again need to account for the

fact that when two groups have at least one overlapping player their outcomes correlate. In

principle, we might expect two groups with more than one overlapping player, such as 1,2,3

4 When the sample size is sufficiently small such that either σ2 and ϕ are poorly estimated or the normal approximation fails, a bootstrap is unlikely to improve inference since it relies on similar asymptotic assumptions.

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and 1,2,4, to have a higher covariance than two groups with just a single overlapping player,

such as 1,2,3 and 1,4,5. However, it becomes cumbersome to keep track of the k – 1

different types of covariances, so we consider the average covariance between any two groups

that overlap by at least one player:

ϕ ≡ Ecov(yi, yj)| pi ∩ pj ≠ ∅ and pi ≠ pj (2-5)

where the expectation is taken with respect to the relative frequency with which each possible

combination of repeated players occurs.

Using this notation, the variance of the estimator is:

var µ( )= 1 J( )2 var yj( )j=1

J

∑ + cov yj , yi( )i≠ j∑

j=1

J

∑⎡

⎣⎢

⎤

⎦⎥

= n−1 Anσ2 + 1− An( )kϕ⎡⎣ ⎤⎦

(2-6)

where An = n/J. Recalling that J is of order nk, it is relatively straightforward to demonstrate that A1 = 1, ∂ An/∂n < 0 and . Thus, nlim

n→∞An = 0 ⋅ var µ( ) converges to kϕ/n and its asymptotic

efficiency relative to the baseline estimator is kϕ/σ2. Note that the two-player game of the

previous section is a special case of a k-player symmetric game for which An = (2n – 1)-1.

2.3 Fully Asymmetric K-Player Games

The generalization from k–player symmetric games to k-player asymmetric games is reasonably

straightforward. We assume that the game is fully asymmetric, in the sense that each of the k

different players enters the outcome function in a unique way. We will relax this assumption in

section 2.4.

Let Si be the set of individuals who played in the ith player role. Then, the set of all

possible groups of players becomes: P = (i1, … , ik)| ij ∈ Sj ∀j ∈ 1, … , k. There are J = nk

elements in P. Under this new definition of P, (2-3) and (2-6) give the recombinant estimator and

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its variance, respectively. Thus, the efficiency of this estimator relative to the baseline estimator

remains kϕ/σ2.

Note that treating a symmetric game as an asymmetric game, which reduces the number

of allowable combinations, leaves the asymptotic efficiency of the estimator unchanged (the

covariance ϕ remains unchanged). As illustrated in Section 6 on computational methods, this

result occurs because the limit on the information from player i is the expected value of the

outcome given that player i is in the game. As the number of recombinations considered

increases, the estimate of this conditional expectation improves, but the number of such

conditional expectations remains fixed. As the sample size gets large, the conditional expectation

for each player is essentially known under either a symmetric or asymmetric treatment, leaving

the standard error of the estimator to be determined by the variability across these conditional

expectations.

2.4 General K-Player Games

Some games of interest have both symmetry and asymmetry between players. For example, in a

sealed-bid double-auction call market with b buyers and s sellers, the buyers are all symmetric to

each other (they each have the same strategy space), and the sellers are all symmetric to each

other, but there is asymmetry between buyers and sellers. Such a game has symmetry within

player roles, but asymmetry between player roles.

Let t be the number of player roles and ti be the number of players of type i in each group,

e.g., t = 2, t1 = b and t2 = s in the double-auction example above. As before, let Si be the set of

individuals who played in the ith player role, e.g., S1 contains all of the buyers in the above

example. Then, the set of all possible groups of players becomes:

P = i11,K , i1t1( ),K , it1,K , ittt( ) i j* ∈Sj ∀j ∈ 1,K , t .

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Under this new definition of P, (2-3) and (2-6) give the recombinant estimator and its variance,

respectively. Again, the efficiency of this estimator relative to the baseline estimator is kϕ/σ2.

3. Theoretical Results on Recombinant Estimation

In this section, we present two propositions. The first proposition demonstrates that the

recombinant estimator is weakly more efficient than the baseline estimator. The second

proposition describes under what circumstances the recombinant estimator is strictly more

efficient than the baseline estimator. All proofs are in Appendix A.

By proving that the average covariance between group outcomes which share

overlapping strategies is less than one kth of the variance of the group outcome function,

Proposition 1 shows that the recombinant estimator is weakly more efficient than the baseline

estimator.

Proposition 1: The recombinant estimator is weakly more efficient than the baseline estimator.

Furthermore, the proof of Proposition 1 illustrates under what conditions the recombinant

estimator is strictly more efficient than the baseline estimator. In particular, it is sufficient that

the outcome function is not an additively separable function of the strategies, i.e.

yj = g xpj1

,K , xpjk( )≠ g1 xpj1( )+K + gk xpjk( ).5

Proposition 2: The recombinant estimator is strictly more efficient than the baseline estimator if the outcome function is not additively separable in the strategies.

Proposition 2 indicates that recombinant estimation provides strict efficiency gains for

many different group outcome functions. The only exception is where the outcome is a linearly

separable function of the players’ strategies. To illustrate an example of this exception, suppose

one is interested in the overall economic efficiency (relative to the Pareto optimum) that can be 5 If recombination across player roles is feasible, then asymmetry across player roles is also sufficient for the recombinant estimator to be strictly more efficient, i.e. gi not equal to gj for at least one pair of i and j.

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obtained in a public-good-contribution game. The game might involve k players each endowed

with 100 units of money, which they can either keep or contribute to the public good. When

player i contributes xi to the public good, that money gets doubled and then divided equally

among the players for their payoffs. In this game, the xi s are the strategies, and the group

outcome is the total payoff to all players. The total payoff is equal to the total amount of public

good provided plus the total amount of private good kept by the players, which turns out to be

100k + Σxi. Since this is linear in the strategies, there is no econometric efficiency gain to be had.

Intuitively, since the group outcome just involves summing up the individual outcomes, the

recombinant estimator turns out to be numerically equivalent to the baseline estimator – it

includes each term in the sum multiple times, but the average turns out to be identical. In the next

section, we demonstrate two examples where the outcome function is nonlinear in the strategies,

and compute the gains that occur.

4. Applied Examples

We present two examples of recombinant estimation. First, we estimate expected revenues and

the allocation of goods in multi-unit auctions, from experiments by List and Lucking-Reiley

(2000). Appendix B provides step-by-step details for the implementation of the recombinant

estimator for this example. Second, we consider expected earnings of players in an ultimatum

bargaining game, from experiments by Mitzkewitz and Nagel (1993).

4.1 Auctions

List and Lucking-Reiley (2000, henceforth LLR) compared two different sealed-bid auction

formats in an experiment at a sportscard trading show. Each of the 328 different bidders

participated in a single two-bidder, two-unit auction for sportscards, with the items’ book values

ranging from $3 to $70. Half the bidders bid in auctions with the uniform-price highest-rejected-

bid price rule, while the other half bid in auctions with the generalized Vickrey price rule. Each

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auction offered a pair of identical sportscards, and each bidder had the opportunity to submit two

bids (one for each of the units of the good). After collecting the bid sheets from individuals over

a period of several hours, the auctioneer randomly paired up individuals from the same type of

auction in order to compute the results. Bidders were asked to return to the same booth at an

appointed time in order to learn the results of the auction and conduct the transaction for any

cards they had won.

LLR had four main results. First, second-unit bids were lower in the uniform-price

treatment than in the Vickrey treatment, as predicted by Nash equilibrium theory. Second, first-

unit bids were higher in the uniform-price treatment than in the Vickrey treatment, an effect

unpredicted by theory. Third, the allocations of goods were significantly different across auction

formats with the uniform-price treatment resulting in more split allocations relative to the

Vickrey treatment. Fourth, there was no statistically significant difference in revenues across

auction formats. LLR’s first two results involved analyses of the mean levels of bids. Since bids

are individual strategies rather than group outcomes, recombinant estimation cannot improve

econometric efficiency for mean bid levels. However, the last two results involve quantities

(revenues and allocations) that are group outcomes. Since each bidder made his bids

independently, we can use the recombination techniques of the present paper to provide

improved estimates of the mean revenues and the frequency of split allocations for each auction

format.

Tables 1 and 2 compare the baseline, non-recombinant estimates from LLR with

recombinant estimates from the same data. Table 1 presents results for the proportion of split

allocations under each format. There are results for ten different auction treatments: five Vickrey

and five uniform-price (the five different treatments involved different goods and different types

of bidders, as shown in the rows of the table). Moving from the baseline to the recombinant

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estimator, the point estimates change by moderate amounts either up or down, and the standard

errors uniformly become smaller. The estimated variance of the recombinant estimator is 44 to

68 percent smaller than that of the baseline estimator, averaging a 57 percent improvement. This

efficiency improvement substantially increases the power of t-tests of the hypothesis that

Vickrey and uniform-price auctions produce equivalent allocations; for example, the two-tailed

p-value for this test in the first treatment (Sanders 1989 card auctioned to Nondealers) decreases

from 0.034 with the baseline estimator to 0.0001 with the recombinant estimator. This

improvement in efficiency is approximately equivalent to a threefold increase in the sample size!

Table 2 presents similar results for auction revenues. Again, the standard errors decrease

when moving from the baseline to the recombinant estimator. Overall, the estimated variance of

the recombinant estimator is 26 percent smaller than that of the baseline estimator, which is

about equivalent to a 35 percent increase in the sample size. In general, the difference in

revenues across auction formats remains statistically insignificant, but the recombinant estimator

provides marginal evidence (p-value of 0.06) that the Vickrey format produces lower revenues

for the Jordan 1989 card.

4.2 Bargaining

Mitzkewitz and Nagel (1993) analyze the ultimatum game. Though a bargaining game does not

typically involve simultaneous moves, Mitzkewitz and Nagel use the “strategy method” to

collect data on complete contingent strategies, which converts the extensive-form, sequential-

move bargaining game into a normal-form, simultaneous-move game. The basic idea of the

ultimatum game is that Player A makes a take-it-or-leave-it offer to split a cash-valued “cake”

with Player B. Player B either accepts the offer so that both players earn the indicated amounts,

or rejects the offer and both players receive nothing. We consider the “demand form” of the

game developed by Mitzkewitz and Nagel, in which the cake size is known to Player A but

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unknown to Player B. All Player B knows is the size of Player A’s demand, but not the size of

the cake or the relative fraction A has demanded. The specific rules of the game are as follows:

1) The cake may be one of six sizes: 1, 2, 3, 4, 5, or 6 Taler, a fictitious currency worth approximately 70 cents.

2) Player A commits to an offer to player B for each of the six possible cake sizes. Offers must be an element of the set 0, 0.5, 1.0,K , 5.5, 6.0 and cannot exceed the size of the cake.

3) Meanwhile, player B, with no knowledge of the cake size, decides which of the possible demands from zero to six she will accept, and which she will reject.

4) After both players have committed to their strategies, a die is rolled to determine the actual size of the cake. Given the cake size, player A’s demand is revealed to player B, who either accepts or rejects it according to her previously elicited strategy. If accepted, player A receives what he demanded and player B receives the remainder of the cake. If rejected, both players receive zero.

This setting fits well the estimation technique of this paper. In fact, Mitzkewitz and Nagel

themselves computed recombinant estimates of expected profits in their analysis (pg. 179).

However, they did not provide standard errors of their estimates, nor did they evaluate the

econometric efficiency of the procedure, so we extend their work in this direction. Table 3

contains the recombinant estimates,6 their associated standard errors, and the relative efficiency

gain of the recombinant estimator.7

This estimation produces two interesting results. First, recombinant estimation yields a 16

to 66 percent efficiency gain in estimating the expected profits of players A and B. The greater

gains are realized at the higher cake sizes (where the strategy space is relatively rich). In general,

the gains to recombinant estimation are roughly proportional to the variance in players’

6 The difference in the point estimates presented here from those in Mitzkewitz and Nagel is attributable to the fact that Mitzkewitz and Nagel computed their estimates by hand. 7 The experiment involved eight repeated trials with each player facing a different rival each round. Following Mizkewitz and Nagel, we recombine not just across individuals within a round, but also across these rounds. We correct the standard errors to account for the possibility of a positive correlation between strategies submitted in different rounds by the same bidder, which is distinct from the positive covariance already assumed to exist between observations using the strategy of the same player in the same round. For ease of exposition, we omit an explicit discussion of the equations used for this purpose, but they are a straightforward generalization of the equations already presented.

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strategies. To be exact, the gain is proportional to the variance in the outcome function induced

by the variance in players' strategies. Second, the estimated standard errors allow us to test an

interesting hypothesis: whether the expected profit to player A is monotonically increasing in the

cake size. Mitzkewitz and Nagel note that the estimated expected payoff to player A takes a

maximum at cake size four and decreases from that point onward, so the point estimates look as

if player A’s profits are not monotonic in the cake size. However, they were unable to compute a

formal test of this hypothesis, because they did not know the distribution of their estimates. The

estimated expected revenue decreases by approximately 0.128 from a cake of size four to one of

size six. The t-statistic for this change is 1.88, so we are able to reject at the 5% significance

level the hypothesis that type-A players’ profits are monotonically increasing in the cake size.

Furthermore, the t-statistic associated with the baseline estimator is only 1.13, which would be

insufficient evidence to reject the null hypothesis. Thus, in this example the recombinant

estimation technique generates a formal rejection of the null hypothesis where standard

estimation techniques did not.

5. Implications for Experimental Design

Our results show that recombinant estimation can produce large improvements in efficiency for

estimating group outcomes from experimental data on auctions and other simultaneous-move

games. We find three key implications of our work for experimental design. First, in dynamic

games, eliciting complete contingent strategies maximizes the returns to recombinant estimation.

Second, in order to recombine across groups of players, all group-level parameters must be held

constant for each group, though individual-level parameters may vary. Third, in order to use

recombinant estimation in an experiment with repeated trials, it is important to have an

experimental design that minimizes repeated contact between the same players. Each of these

implications is discussed in more detail below.

17

5.1 Complete Strategies

Consider a two-player, dynamic game in which player A moves first, then player B responds.

Suppose player A has five potential moves labeled one through five. Consider a player in

position A who chose option two. This player may only be recombined with players in position

B whose opponent also chose option two. Alternatively, suppose the experimenter had complete

contingent strategies from each player in position B. Then, each player in position A could be

recombined with all players from position B. To attain complete contingent strategies and take

full advantage of recombinant estimation, experimenters may wish to consider the use of the

strategy method (as done in Mitzkewitz and Nagel, 1993).

5.2 Consistency Across Sessions

The experimenter needs to make sure that experimental conditions are held constant across

experimental sessions in a way that allows for recombination, a technique known as a “yoked”

design in experimental psychology. When using a yoked design, all sessions of the experiment

will appear ex-ante identical to every player. Thus, switching a player from session i to session j

will not alter her strategy. Typically changing an experimental design into a yoked design

requires minimal effort. An example may be helpful.

Kagel et al. (1987) tested the theory of auctions with affiliated private values. In each

experimental auction, they used the following procedure. First, they chose a locational parameter

x0 randomly from a uniform distribution on [$25, $125]. Second, they chose a set of six private

values xi for the six bidders, with each xi drawn independently from the uniform distribution on

[x0 – ε, x0 + ε], where ε was a constant equal to either $6, $12, or $24, depending on the round of

the experiment. Each bidder learned his private value xi, but did not necessarily learn the value of

x0, so he might remain uncertain about the distribution from which his value came. This

generated “affiliation” between bidders’ values, as described by Milgrom and Weber (1982).

18

That is, when a bidder learns that his value is high, he also learns that other bidders’ values are

likely to be relatively high.

To keep the discussion short, we consider only the first auction of their experiment,

observed in each of seven separate sessions. If the design were yoked, there would be over 5.2

million different hypothetical revenue observations (combinations of 42 different subjects taken

six at a time), instead of the mere seven observed outcomes. The exact amount of efficiency

improvement would depend on the actual distribution of the data, but it is likely to be quite large.

Unfortunately, the locational parameter x0 makes it impossible to recombine bidders. The value

of x0 was drawn independently across sessions, leaving no two groups of six bidders with the

same value. In order to use the recombinant estimator, one would have to hold the locational

parameter fixed across sessions. Additionally, the range-of-values parameter ε, would need to be

held constant across sessions. Note that it is not necessary to use the same set of individual

values xi in each experimental session, because the bids from bidders with different individual

valuations could still be recombined, so long as the group-level parameters are the same. In fact,

it would be preferable to generate different random bidder values across sessions, because this

would create a better representation of the full population to be studied: all bidders who might

possibly have bid in an auction with the particular values of x0 and ε.

5.3 Learning

A key assumption underlying recombinant estimation is that strategies in different sessions are

all independent draws from the same distribution. For example, in games with repeated play, in

order to combine one individual’s strategy in round 2 with another individual’s strategy in round

10, one must believe that there is no systematic change in play over time. Although there exists a

large literature on testing the hypothesis that two samples are drawn from the same distribution,

the sample sizes in most experiments are sufficiently small such that the power of these tests

19

would be very low. In lieu of focusing on these formal tests, we only consider two deviations

from this assumption: learning about the game, and learning about one’s opponents.

Suppose players learn about the game over successive rounds of play. For example,

bidders gradually learn to bid lower. Under this scenario, bids in round 2 are drawn from a

different distribution than bids in round 10. Thus, it is inappropriate to assume that a player in

round 10 would have made the same bid if she were returned to round 2. In other words, one

should not recombine across rounds, but only across sessions within the same round.

Another complication is the possibility of learning not just about the game but also about

one’s rival players: for example, one group of players learns to submit high bids because they

played against a rival who bids unusually high in early rounds. If subjects do learn about their

specific group of rivals in repeated play, then recombining across groups in any round after the

first would be inappropriate. Although it might be possible to construct a recombinant estimator

that also controls for group effects of this type, this task is beyond the scope of the present paper.

In order to avoid learning about one’s rivals in an experiment with repeated trials, the

experimenter might wish to choose an experimental design that minimizes the possibility for this

type of learning. For example, Mitzkewitz and Nagel’s experimental design ensured that over the

eight rounds of the experiment, no player ever played against the same rival twice. Thus, in

section 4.2, we feel relatively sanguine about recombining within rounds for that experiment. An

even stronger insurance would be the “zipper design” sometimes used in experiments, where in

repeated rounds a player never encounters anyone who might have been influenced by that

player previously. (That is, one never plays one’s previous rivals again, nor one’s rivals’ rivals,

nor one’s rivals’ rivals’ rivals, and so on.)

20

6. Computational Methods

Even a relatively small number of observations (seven groups of six subjects each) can yield a

very large number of combinations. This explosive growth in the number of combinations, both

with respect to the number of players in a game and the total number of players, could become

computationally burdensome. Fortunately, the vast majority of the efficiency gain from

recombinant estimation can be obtained from a relatively small fraction of those possible

combinations. We show below that 100 combinations per player are usually sufficient. We start

with the computation of point estimates and finish with the estimation of standard errors.

6.1 Computationally Convenient Point Estimates

Another way to express the recombinant estimator is via iterated expectations. Recall that P

represents the set of all hypothetical groupings of the observed strategies. Let Pi be the subset of

P containing only those groups which include player i. That is, Pi = pj ∈ P|i ∈ pj. Let Ji

represent the number of elements in Pi. Now define Ey|i as the conditional expectation of y

restricted to groups containing player i. An estimate of this expectation is:

E y i = 1Ji

⎛⎝⎜

⎞⎠⎟

yjj :pj ∈Pi

∑ . (6-1)

Express the recombinant estimator as the average of these estimated conditional

expectation over all players i:

µ =1m

⎛⎝⎜

⎞⎠⎟

E y ii=1

m

∑ . (6-2)

As written in (6-2), each group of players is in this summation exactly k times. However,

counting every combination k times does not alter the average value.

Consider using r ≤ Ji different combinations involving each player to estimate Ey|i.

Then, (6-2) becomes

21

µr =1m

⎛⎝⎜

⎞⎠⎟

1r

⎛⎝⎜

⎞⎠⎟

yjj :pj ∈Ri

∑i=1

m

∑ . (6-3)

where Ri contains r random draws from Pi. When r = 1, µr reduces to the baseline estimator,

which involves no recombinations at all. Clearly, r = 1 gives a poor estimate of Ey|i. Including

more recombinations allows the researcher to better estimate Ey|i for each player. However,

the gain to recombining is limited by the fact that all it provides is a better estimate of Ey|i; it

does not increase the number of conditional expectations considered. Therefore, the variance of

the estimator is bounded below by the variance of the estimator in which all of the conditional

expectations are known:

µ*

=1m

⎛⎝⎜

⎞⎠⎟

E y ii=1

m

∑ (6-4)

Thus, once r is large enough such that the variance of ˆrµ is close to this lower bound, then there

is little gain to considering additional recombinations of the data.

To get an idea of how large r needs to be, consider the variance of µr . The probability

that pj and pi share at least one player converges to k2/m from above, as n goes to infinity.8 So

var µ( )≈ σ 2

mr+

k2

mϕ =

σ 2

krn+

kϕn

(6-5)

where ≈ denotes approximate equality when the sample size is large. When all possible

combinations involving a particular player are used, then r is of order mk-1, resulting in the first

term in (6-5) being of lower order. However, if r is fixed, then the first term goes to zero at the

same rate as the second, so we have a less efficient estimator. In particular, the asymptotic

efficiency of the recombinant estimator with r fixed relative to the full-blown recombinant

estimator is 1 + σ2/(rk2ϕ).

8 The exact probability that two randomly selected games share at least one player is

1− m − k − i( ) m − i( )i=0

k∏ .

22

The exact gain from increasing the number of combinations considered is a function of

the ratio between ϕ and σ2. Table 4 displays the relative efficiencies of the baseline, recombinant

and restricted estimators with r fixed at 50, 100 and 500. The table illustrates two points. First, as

the covariance across combinations with at least one shared player decreases relative to the

variance of the group outcome function, the potential gain from recombining data increases.

Second, since the gain to recombining data is great when the covariance is small, there is a

greater loss associated with restricting the estimator to consider r recombinations when the

covariance is small, as illustrated by the final column of the above table. In particular, consider

the case where there are two players in the game and the covariance is 0.01. Here, restricting the

estimator to 100 recombinations results in a 25% efficiency loss relative to the fully recombinant

estimator. However, we still capture 97.5% of the 98% efficiency gain relative to the baseline

estimator.

6.2 Computationally Convenient Standard-Error Estimates

Computation of standard errors for the recombinant estimator requires separately estimating both

the variance of the outcome and the average covariance when there is at least one shared player,

σ2 and ϕ, respectively. Both σ2 and ϕ can themselves be estimated via recombinant estimation,

because they are themselves means of different group outcomes. To see this result, define a new

outcome as the squared deviation from the average value of the old outcome, i.e.

s j = yj − µ( 2) . (6-6)

Then the average value of this new outcome is the sample variance of the original outcome

variable y, which makes it an estimate of the population variance σ2. Since we have now posed

the estimation of σ2 as the estimation of a mean value, we can use the recombinant technique to

estimate σ2 more efficiently, computing this mean across all possible k-player groups.

Equivalently, one generates a recombinant estimate of σ2 by generating the same J different

23

hypothetical outcomes used to compute µ , and taking their sample variance (whereas µ was

their sample mean).

Recombinant estimation can also be used to estimate ϕ. To see this, let Ω = pi, pj ∈ P| pi

∩ pj ≠ ∅. Define a group of players as ωij = pi, pj ∈ Ω and the outcome as:

hij = yi − µ( ) yj − µ( ). (6-7)

Then ϕ equals the expected value of this new outcome, restricted to observations where at least

one player overlaps between groups i and j. Thus, ϕ is the expected value of a new type of group

outcome in which the group size is 2k – 1 players (two different groups of size k which are

constrained to overlap by at least one player). Since ϕ is the average value of a new type of

group outcome, it can be estimated with the recombinant technique. In practice, to estimate ϕ

one generates two columns of data, each row of which contains two different hypothetical

outcomes that overlap by at least one player. After generating all such possible pairs of

outcomes, one takes the sample covariance between the two columns in order to estimate ϕ.

In general, recombinant estimation of ϕ is the most computationally expensive task we

consider, because the number of such pairs of outcomes is of order n2k-1, rather than of order nk

as required to estimate µ or σ2. Thus, in practical computation of standard errors, it is essential to

restrict attention to a subset of the possible combinations as was discussed in Section 6.1. In

practice, we recommend generating at least 100 random pairs of outcomes overlapping in player

1, 100 pairs overlapping in player 2, and so on up to player m then computing the sample

covariance across all such pairs.

Because our estimate of σ2 comes from recombinant estimation of an outcome (see (6-6))

that is nonlinear in the strategies, this estimate is strictly more efficient (see Proposition 2) than

the standard non-recombinant estimate of σ2 taken across only observed groups. If this result

leads to a more accurate estimate of the standard error (recall that we have to estimate ϕ, as well

24

as σ2), then we will have even greater power in hypothesis testing. This result, although not fully

developed here, is analogous to the power in a t-test. Recall that in a t-test, increasing the degrees

of freedom (while holding the point estimate and the population standard deviation constant) also

increases the power of the test. However, as with a t-test, the gain in power to increasing the

precision of the estimate of the variance is decreasing in the precision of the original estimate.

Thus, when the sample size is large, the recombinant and the simple estimators of the variance

will produce tests of the same power (just as a t-test with 500 degrees of freedom has essentially

the same power as one with 1000 degrees of freedom).

7. Concluding Remarks

We have presented a new econometric estimator for use in applications where the quantity of

interest is a group outcome based upon the individual strategies of the players. Our recombinant

estimator considers all possible hypothetical groupings, yielding considerable efficiency

improvements relative to the standard estimator that considers only the observed groupings. In

the applied examples we have examined so far, we find efficiency improvements equivalent to

increases in the sample size of between 25 and 200 percent. Even greater efficiency

improvements are possible in principle, but the exact amount of improvement depends on both

the functional form of the group outcome function and on the distribution of the population of

individual strategies. We have shown our estimator guarantees some efficiency improvement so

long as the group outcome function is not additively separable over the individual strategies.

Experiments on auctions and other normal-form games provide the most straightforward

applications of our technique. Recombinant estimation can yield lower-variance estimates of

such quantities as auction revenues, market efficiency, or the fraction of the time players achieve

the Nash equilibrium in a matrix game. This technique may also help rescue data that might

otherwise have become unusable when an experiment goes wrong. For example, suppose that in

25

a twelve-person, simultaneous-move market experiment, one of the twelve participants leaves

before submitting a strategy. The market efficiency for that group of twelve subjects cannot be

calculated, because one can’t find the actual allocation when one of the strategies is missing.

However, the other eleven players all submitted valid strategies, so rather than throwing out

these observations, one may recombine them with the data from other sessions in order to yield

market efficiency estimates.9

The estimator also holds some promise beyond simultaneous-move game experiments,

and even beyond the field of economics. For example, consider a medical statistician who wishes

to estimate the average ratio between healing time with drug A and healing time with drug B.

Half the patients in the study received drug A, while the other half received drug B. This

situation is isomorphic to the estimation of a two-player game, where the strategies are healing

times and the outcome is the ratio between the healing times of one patient and another. The

recombinant estimator considers all possible pairings of A-types with B-types, producing an

estimate (and a standard error) for this ratio. The technique might be used in any statistical

situation that involves matching or grouping of observations.

Future research might be able to generalize our results, for example by exploring the

properties of recombinant estimation when the econometrician expects within-group fixed effects

in repeated observations of the same group, or by finding additional applications for recombinant

estimation. Most importantly, we hope that our work, with its discussions of the practical aspects

of computation and of experimental design for use with this technique, will enable other

researchers to improve the efficiency of their estimates of behavior in simultaneous-move games.

We hope to make this easy for other researchers by providing the software described in footnote

1.

9 We are grateful to Yan Chen for bringing this application to our attention.

26

8. Appendix A

This appendix contains a proof of Proposition 1, that the recombinant estimator is weakly

asymptotically more efficient than the baseline estimator. Proposition 2, that the recombinant

estimator is strictly more efficient when the group outcome function is not additively separable

in players' strategies, is proved as a special case. One lemma and two properties of functions of

independent random variables are invoked in the proof. Maintaining the notation of section 2, let

ϕ ≡ Ecov(yi, yj)| pi ∩ pj ≠ ∅ and pi ≠ pj.

The lemma states that the average covariance between groups with at least one common player

converges to the average covariance between groups with exactly one common player. More

precisely,

Lemma 1: limn→∞

ϕ =ψ where ψ ≡ Ecov(yi, yj)| ∃ exactly one pair (t, τ): pit ≠ piτ.

Proof: The number of pairs yi, yj with exactly r repeated players is of order n2k-r. Therefore, the proportion of the population of pairs with shared players that share exactly one player converges to one as n approaches infinity. QED

The two properties of functions of independent random variables are as follows:

1) Let x be a random k-dimensional vector where xi is independent of xj for all i ≠ j. Then, for all f(x): Rk → R, f can be expressed as f(x) = f1(x1) + … + fk(xk) + δ(x) where xi is correlated with fi(xi) and uncorrelated with all other terms in f. Furthermore,

with equality if and only if δ(x) = 0.

corr f x( ), fi xi( )( )⎡⎣ ⎤⎦2

i=1

k

∑ ≤ 1

2) Let x be a random variable, y and z be k-dimensional random vectors where x, y and z are independent of each other. Then, for all f(x): Rk+1 → R, corr f x, y( ), f x, z( )( )⎡⎣ ⎤⎦

2= corr f x, y( ), fx x( )( )⎡⎣ ⎤⎦

2corr f x, z( ), fx x( )( )⎡⎣ ⎤⎦

2

where fx(x) is defined as in property (1).

27

Armed with these three results, we are prepared to prove Proposition 1, that the

recombinant estimator is weakly more efficient than the standard estimator. In the proof, we

point out where the lack of additive separability in the group outcome function leads to a strictly

more efficient estimator, proving Proposition 2. Finally, we state Proposition 1 inclusive of all

the assumptions in the paper, noting that the relative efficiency of the recombinant estimator is

kϕ/σ2.

PROPOSITION 1: Let xi be independently and identically distributed random variables and . Then, kϕ is asymptotically less than or equal to σ

j p j1 pjky = g x ,K , x( ): Rk → R 2.

Proof: By lemma 1, it is sufficient to prove that the claim holds true when we restrict attention to

pairs of outcomes that share exactly one player.

Recall that each player is only repeated in the role in which she is observed and that the

game is symmetric within each player role. Therefore, without loss of generality, assume that if a

player participates in both yi and yj, she is in the same position in both games. There are k player

positions, so there are k equally likely types of terms that share exactly one player with yi in this

manner: for t = 1 to k, where indicates that a player from the tyjt yj

t th position has been repeated

in game j. (Each type is equally likely because the set of players is restricted to unique

combinations where the order of players within a player role does not matter.) Note that is

independent of

tjy

jyτ for all t ≠ τ since the common player with group i differs and the remaining

players are drawn at random.

Let ρij , t = corr yi , yjt( ). By Property (2),

ρij , t2 = ρggt

2 ρggt

2 = ρggt

4 .

Therefore, the sum of the correlations over the k-player positions is

ρij , tt=1

k

∑ = ρij , t2

t=1

k

∑ = ρggt

2

t=1

k

∑ ≤ 1

28

where the final inequality is a direct application of Property (1). Note that the final inequality above becomes strict if yj = g xpj1

,K , xpjk( ) cannot be

expressed as an additively separable function of the elements of x, i.e. δ(x) ≠ 0 in the

decomposition in property (1). Furthermore, a strict inequality will carry through to a strict

inequality in the statement of Proposition 1. This means a strictly more efficient estimator, and

thus proves Proposition 2.

Returning to the proof of Proposition 1, algebraic manipulation of the above inequality

yields the desired result, i.e.

1≥ ρij , tt=1

k

∑ =cov yi , yj pit = pjt( )

var yi( )var yj( )t=1

k

∑ =cov yi , yj pit = pjt( )

var y( )t=1

k

∑

⇒ cov yi , yj pit = pjt( )t=1

k

∑ ≤ var y( )

⇒1k

cov yi , yj pit = pjt( )t=1

k

∑ ≤var y( )

k

⇒ϕ ≤var y( )

k

Since limn→∞

ϕ =1k

cov yi , yj pit = pjt( )t=1

k

∑ by lemma 1 and the fact that each type of covariance in

the summation is equally likely.

29

9. Appendix B

Since the ten auction treatments considered in LLR are all two-player symmetric games, it is

sufficient to consider the first treatment, Vickrey auctions of the Barry Sanders card. The data

from this experiment are listed in the following table.

First Second First Second First SecondBid Bid Bid Bid Bid Bid

Player 1 40 30 Player 13 60 60 Player 25 85 68Player 2 60 45 Player 14 45 30 Player 26 20 20





Player 11 75 25 Player 23 100 50Player 12 55 10 Player 24 78 48

Bids from the Barry Sanders 1989 Vickrey AuctionsEach player's bids are listed and players are paired as they were observed in the data.

Furthermore, we restrict our attention to the revenue generated by these auctions, which is the

sum of the lowest two bids in each auction.

There were 17 auctions in this format with two bidders in each auction, yielding 34

players. P, the set of all possible combinations of two players, has J = 34(34 – 1)/2 = 561 unique

elements. The point estimate is computed by taking the average of the revenue over these 561

auctions, which is equivalent to computing the double summation below.

33 34

,1 1 1

1 1ˆ 49.17561 561

J

i i ji i j i

y yµ= = = +

= = =∑ ∑∑ .

In the above equation, yi, j is the revenue generated by the auction involving players i and j.

30

The 561 outcomes, y1, … , y561, can be used to estimate both the variance of a single

game and the covariance between two games with one shared player. The variance is

σ 2 =1

561−1( )yi − µ( )2

i=1

J

∑ =1

561−1( )yi

2

i=1

J

∑ − 561µ2⎡

⎣⎢

⎤

⎦⎥ = 736.14

where the second representation is computationally more convenient. The covariance is

( ) ( )( )

( )

1

1 11

1 1

ˆ ˆ1ˆ 296.56

1

J J

i k i ki k i

J J

i ki k i

p p y y

p p

µ µϕ

−

= = +−

= = +

∩ ≠ ∅ − −⎡ ⎤⎣ ⎦= =

∩ ≠ ∅

∑ ∑

∑∑

where 1(pi ∩ pk ≠ ∅) is equal to one if games i and k share a common player and zero otherwise.

Then, 2σ and ϕ are substituted into (2-4) for σ2 and ϕ to yield

var µ( )= 2m m −1( )

σ 2 +4 m − 2( )m m −1( )

ϕ =σ 2 + 64ϕ

561=

736.14 + 64 296.56( )561

= 35.14 ,

which implies that the standard error of the mean is 5.93.

Finally, the recombinant estimate of the variance was used to estimate the variance of the

baseline estimator (736.14/17 = 43.3).

For more information, download the Microsoft Excel workbook that implements our

estimator (see footnote 1), which provides the data from this application as an example.

31

32

References

Friedman, D., 1996. Equilibrium in Evolutionary Games: Some Experimental Results. The

Economic Journal 106, 1 – 25.

Kagel, J. H., Harstad, R. M., Levin, D., 1987. Information Impact and Allocation Rules in

Auctions with Affiliated Private Values: A Laboratory Study. Econometrica 55, 1275 –

1304.

List, J., Lucking-Reiley, D., 2000. Demand Reduction in Multi-Unit Auctions: Evidence from a

Sportscard Field Experiment. American Economic Review 90, 961 – 972.

Mehta, J., Starmer, C., Sugden, R., 1994. The Nature of Salience: An Experimental Investigation

of Pure Coordination Games. American Economic Review 84, 658 – 673.

Milgrom, P. R., Weber, R. J., 1982. A Theory of Auctions and Competitive Bidding.

Econometrica 50, 1089 – 1122.

Mitzkewitz, M., Nagel, R., 1993. Experimental Results on Ultimatum Games with Incomplete

Information. International Journal of Game Theory 22, 171 – 198.

Moir, R., 1998. A Monte Carlo Analysis of the Fisher Randomization Technique: Reviving

Randomization for Experimental Economists. Experimental-Economics 1, 87 – 100.

Table 1 Proportion of Auctions with the Two Units Split Between the Two Bidders:

A Comparison of the Baseline and Recombinant Estimators (Standard Errors in Parentheses; P-Values in Double Parentheses)

Vickrey Uniform Vickrey Minus Uniform Trading Card LLR Recombinant LLR Recombinant LLR RecombinantBarry Sanders 1989 Score; BV = $70 0.56 0.53 0.85 0.87 -0.29 -0.34Nondealers

(0.12)

(0.07) (0.08)

(0.06) (0.14) (0.09)((0.034)) ((0.000))

Cal Ripken 1982 Topps; BV = $70

0.23 0.30 0.87 0.88 -0.63 -0.58Dealers

(0.10)

(0.06) (0.08)

(0.05) (0.13) (0.08)((0.000)) ((0.000))

Jordan 1989 Hoops; BV=$3

0.32 0.34 0.54 0.52 -0.22 -0.17Nondealers

(0.09)

(0.05) (0.09)

(0.06) (0.13) (0.08)((0.079)) ((0.036))

Montana 1982 Topps; BV = $3

0.57 0.50 0.45 0.56 0.12 -0.06Dealers

(0.14)

(0.09) (0.15)

(0.10) (0.20) (0.13)((0.565)) ((0.675))

Montana 1982 Topps; BV=$3

0.47 0.48 0.53 0.54 -0.07 -0.07Nondealers

(0.11)

(0.07) (0.11)

(0.08) (0.16) (0.11)((0.678)) ((0.562))

33

Table 2 Average Revenue from Vickrey and Uniform Sealed Bid Auctions:

A Comparison of the Baseline and Recombinant Estimators (Standard Errors in Parentheses; P-Values in Double Parentheses)

Vickrey Uniform Vickrey Minus Uniform Trading Card LLR Recombinant LLR Recombinant LLR RecombinantBarry Sanders 1989 Score; BV = $70 51.06 49.17 48.71 45.25 2.35 3.92Nondealers

(6.58)

(5.93) (6.69)

(5.87) (9.39) (8.35)((0.802)) ((0.639))

Cal Ripken 1982 Topps; BV = $70

72.87 73.52 76.13 74.30 -3.27 -0.77Dealers

(6.56)

(5.98) (5.42)

(4.72) (8.51) (7.62)((0.701)) ((0.919))

Jordan 1989 Hoops; BV=$3

1.13 1.09 1.71 1.72 -0.58 -0.63Nondealers

(0.26)

(0.22) (0.29)

(0.25) (0.39) (0.33)((0.142)) ((0.055))

Montana 1982 Topps; BV = $3

2.37 2.16 2.13 2.14 0.24 0.02Dealers

(0.32)

(0.29) (0.44)

(0.36) (0.54) (0.46)((0.659)) ((0.959))

Montana 1982 Topps; BV=$3

0.66 0.65 0.83 0.88 -0.17 -0.23Nondealers

(0.19)

(0.16) (0.28)

(0.22) (0.34) (0.27)((0.613)) ((0.403))

34

Table 3 Mean Expected Payoffs for Demanders and Repliers in the Ultimatum Game

(Standard Errors in Parentheses)

Cake 1 2 4 5 63 Player A

Point Estimate

0.783 1.441 1.874 1.945 1.916 1.816(0.014) (0.031) (0.043) (0.049) (0.059) (0.065)

Efficiency Gain

0.287 0.383 0.555 0.655 0.626 0.633

Player B

Point Estimate

0.167 0.327 0.525 0.910 1.248 1.528(0.012) (0.022) (0.027) (0.034) (0.047) (0.064)

Efficiency Gain 0.164 0.210 0.262 0.388 0.404 0.451

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Table 4

Relative Efficiency of the Computationally Restricted Estimator

Number of Recombinant/ Restricted/ Restricted/

Players

phi/sigma2 Baseline Baseline

Recombinant

50 Recombinations

2 0.250 0.500 0.510 1.0202 0.100 0.200 0.210 1.0502 0.010 0.020 0.030 1.5005 0.100 0.500 0.504 1.0085

0.010 0.050 0.054 1.080

100 Recombinations

2 0.250 0.500 0.505 1.0102 0.100 0.200 0.205 1.0252 0.010 0.020 0.025 1.2505 0.100 0.500 0.502 1.0045

0.010 0.050 0.052 1.040

500 Recombinations

2 0.250 0.500 0.501 1.0022 0.100 0.200 0.201 1.0052 0.010 0.020 0.021 1.0505 0.100 0.500 0.500 1.0015 0.010 0.050 0.050 1.008

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recombinant estimation for normal-form games, with ... · recombinant estimation for normal-form...

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